Calculus Examples

$\int (2 x + 1) {(x - 6)}^{2} d x$

Step 1

Let $u = x - 6$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x - 6$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x - 6$ .

$\frac{d}{d x} [x - 6]$

Step 1.1.2

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 6]$

Step 1.1.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int (2 (u + 6) + 1) u^{2} d u$

Step 2

Expand $(2 (u + 6) + 1) u^{2}$ .

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Step 2.1

Apply the distributive property.

$\int (2 u + 2 \cdot 6 + 1) u^{2} d u$

Step 2.2

Apply the distributive property.

$\int (2 u + 2 \cdot 6) u^{2} + 1 u^{2} d u$

Step 2.3

Apply the distributive property.

$\int 2 u \cdot u^{2} + 2 \cdot 6 u^{2} + 1 u^{2} d u$

Step 2.4

Raise $u$ to the power of $1$ .

$\int 2 (u^{1} u^{2}) + 2 \cdot 6 u^{2} + 1 u^{2} d u$

Step 2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 2 u^{1 + 2} + 2 \cdot 6 u^{2} + 1 u^{2} d u$

Step 2.6

Add $1$ and $2$ .

$\int 2 u^{3} + 2 \cdot 6 u^{2} + 1 u^{2} d u$

Step 2.7

Multiply $2$ by $6$ .

$\int 2 u^{3} + 12 u^{2} + 1 u^{2} d u$

Step 2.8

Multiply $u^{2}$ by $1$ .

$\int 2 u^{3} + 12 u^{2} + u^{2} d u$

Step 2.9

Add $12 u^{2}$ and $u^{2}$ .

$\int 2 u^{3} + 13 u^{2} d u$

Step 3

Split the single integral into multiple integrals.

$\int 2 u^{3} d u + \int 13 u^{2} d u$

Step 4

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$2 \int u^{3} d u + \int 13 u^{2} d u$

Step 5

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$2 (\frac{1}{4} u^{4} + C) + \int 13 u^{2} d u$

Step 6

Since $13$ is constant with respect to $u$ , move $13$ out of the integral.

$2 (\frac{1}{4} u^{4} + C) + 13 \int u^{2} d u$

Step 7

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$2 (\frac{1}{4} u^{4} + C) + 13 (\frac{1}{3} u^{3} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$2 (\frac{1}{4}) u^{4} + 13 (\frac{1}{3}) u^{3} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $2$ and $\frac{1}{4}$ .

$\frac{2}{4} u^{4} + 13 (\frac{1}{3}) u^{3} + C$

Step 8.2.2

Cancel the common factor of $2$ and $4$ .

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Step 8.2.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4} u^{4} + 13 (\frac{1}{3}) u^{3} + C$

Step 8.2.2.2

Cancel the common factors.

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Step 8.2.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2} u^{4} + 13 (\frac{1}{3}) u^{3} + C$

Step 8.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2} u^{4} + 13 (\frac{1}{3}) u^{3} + C$

Step 8.2.2.2.3

Rewrite the expression.

$\frac{1}{2} u^{4} + 13 (\frac{1}{3}) u^{3} + C$

Step 8.2.3

Combine $13$ and $\frac{1}{3}$ .

$\frac{1}{2} u^{4} + \frac{13}{3} u^{3} + C$

Step 9

Replace all occurrences of $u$ with $x - 6$ .

$\frac{1}{2} {(x - 6)}^{4} + \frac{13}{3} {(x - 6)}^{3} + C$